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Time implicit schemes and fast approximations of the Fokker-Planck-Landau equation
by Mohammed Lemou   Luc Mieussens  

Vol. 2 No. 2 (2007) P.533~P.567

ABSTRACT

In this paper, we are concerned with numerical approximations of the Fokker-Planck-Landau equation which is a kinetic model used to describe the evolution of charged particles in a plasma. In this model, the particle interactions (or collisions) are taken into account by a nonlocal and nonlinear di ffusion operator acting on the velocity dependence of the particle distribution function. In a first part of this work, we investigate di fferent strategies to perform efficient time implicit discretisations, while, in the second part, we review various numerical approximations of the collision operator. Both the time discretisation and the approximations of the collision operator are shown to satisfy some important physical properties of conservation and entropy, and to reach the right steady states. Furthermore, various accelerations techniques are used to construct such approximations which would make possible their use in a more realistic setting (inhomogeneous cases). In particular, we combine two new strategies to rapidly and efficiently solve the FPL equation: the fi rst one concerns the time discretisation using time implicit schemes with Krylov solvers, and the second one uses the approximation of the collision operator using the wavelet approximation theory.


KEYWORDS
Kinetic equations, Fokker-Planck-Landau equation, implicit schemes, conservative schemes, Krylov methods, wavelets

MATHEMATICAL SUBJECT CLASSIFICATION 2010
Primary: 82C40, 82D10, 82C80, 65M06, 65Y20, 65F10

MILESTONES

Received: 2004-12-10
Revised : 2005-05-19
Accepted:


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